Problem Set 03

Note: The problem set is due October 2 by 4:00 pm. In case you’re unable to attend the

class, please return directly to me in my office Rutherford 321. If I’m not there slip your

assignment below my door. Please make sure you submit a hard-copy of the assignment as

it is easier for the TA’s to correct!

1. I hang two pulleys of equal size, meaning they have the same radii, from the ceiling.

The wires going through the two pulleys make angles α along a vertical axis, and I suspend

a mass M as shown in the following figure:

At the instant when the angles are α on both sides of the axis, the wires hanging at the

two ends of the pulley are pulled down with a constant velocity of v. Determine with what

velocity will the mass M go up.

2. A set of pulleys are arranged in a way that one pulley is supported by another by a

wire that goes around the other pulley as shown in the following figure:

We now hang a mass M1 on the left pulley and another mass M2 on the other pulley. For

generic values of M1 and M2 the system will start moving. Can you find, for what values

of M1 and M2, the system will be static? Provide quantitative details.

3. As we discussed in the class, if we put unequal masses connected by a wire on the

two sides of an inclined plane the system will start moving, generically along the direction

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of the heavier mass. Let us then consider an arrangement of fourteen spheroids slipped

around a hanging inclined plane in the following fashion:

The bottom droops as a garland, which is in fact in a state of balance. However on one side

of the plane there are four spheroids, while the other side has only two spheroids. Thus

there seems to be an imbalance of force, and from our analysis it would seem that the

system will start moving. If there is no friction, this motion should continue forever, and

we have thus succeeded in constructing a perpetual machine! Is it true? Give quantitative

reasons.

4. A mass M1 is kept on the inclined side of a plane of mass M2 and inclination α in such

a way that the wire connecting the mass M1 runs through a pulley at the top edge of the

inclined plane to a static wall. The configuration is shown in the following figure:

In the class we computed the acceleration of the inclined plane using Lagrangian method.

Using only Newtonian techniques now, show that we get similar answer for the acceleration.

Describe all the forces acting on the individual parts of the system. Assume no friction1.

5. Two masses M1 and M2 are arranged on an inclined plane of inclination α by a wire

going over two pulleys. One of the pulley is arranged at the top end of the inclined plane

1 It is of course interesting to speculate what happens in the presence of friction between the

ground and the inclined plane, but no friction between the mass M1 and the inclined plane; or

vice versa.

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whereas the second pulley hangs on the other end of the wire, and is thus movable. From

the second pulley we hang the mass M2 by a second wire that goes over the second pulley

and is fixed on the inclined plane. Both the pulleys are massless, and the inclined plane

is fixed in such a way the motion of M2 is not impeded. The arrangement is shown in the

following figure:

Now assume that M1 > M2 such that the motion is dominated by M1. Determine the

acceleration with which the mass M2 will go up. Provide quantitative details. You may

use either the Lagrangian or the Newtonian methods.

6. Another arrangement of a movable inclined plane, of mass M1, inclination angle α and

with a system of pulleys, rests on a frictionless surface. The mass M2 is connected by a

wire that runs over the pulley system as shown in the following figure:

When the mass M2 moves, it makes the inclined plane move towards the wall. Determine

the acceleration with which the inclined plane will move. As before, you are free to use

either the Lagrangian or the Newtonian techniques.

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